LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exterior exhaustion bounds for normalized Schrödinger solutions

Statement

On a connected exterior as in the exterior-data definition, let $V \geq 0$ be smooth and let $u \in C^{2} (Σ^{\circ}) \cap C_{l o c}^{1} (Σ)$ solve $(Δ_{h} - V) u = 0$ , vanish on $S$ , and tend uniformly to $c > 0$ as $r \to \infty$ . Then $0 < u \leq c$ in the interior. If $S$ is nonempty, then $u < c$ there.

Facts & Assumptions

Given: The end chart and compact core are those of Weighted exterior elliptic domains and admissible weights. The limit means that for every $ϵ > 0$ , $∣ u - c ∣ < ϵ$ beyond a sufficiently large end sphere. Use compact comparison Compact-domain comparison with a nonnegative potential and the strong minimum alternative Strong minimum principle from the annular barrier.

Proof

Given: The hypotheses and data in the Statement and Facts & Assumptions.

Goal: Establish the conclusions of the Statement.

Proof technique: direct.

1.1given

Truncate the exterior at $r = R$ . For $R$ sufficiently large, $u \geq c / 2 > 0$ on the outer sphere and $u = 0$ on $S$ . Comparison applied to $- u$ gives $u \geq 0$ on the truncation. Applied to $u - (c + ϵ)$ , whose image under $Δ_{h} - V$ is $V (c + ϵ) \geq 0$ , it gives $u \leq c + ϵ$ when the outer sphere satisfies that bound. Every fixed point belongs to all sufficiently large truncations. Taking $R$ large and then $ϵ ↓ 0$ proves $0 \leq u \leq c$ . Components of a truncation meet its boundary; otherwise such a component would be a compact component of the connected exterior separated from its end.

2.1step 1.1givenalgebra∎

The end limit excludes $u \equiv 0$ , so the strong minimum alternative makes $u > 0$ . The nonnegative function $c - u$ satisfies $(Δ_{h} - V) (c - u) = - V c \leq 0$ . If it vanished inside, the same alternative would make $u \equiv c$ . This contradicts $u ∣_{S} = 0$ when $S \neq \emptyset$ . Thus the stated strict upper bound also holds.

Depends on

Used by · 1

Exterior Schrödinger maximum principles, Hopf growth and Harnack

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.